So I have been checking out the support ship, but I am looking at the best build to make in regards to how I spend my CP. For the most part I try to work on fleet, but do things like increased attack power improve my turrets?

One player should not really gimp himself by contributing more than his fair share to the economy.
Put 300 into colonization, and if you hold a wormhole, 100 each into that too. Beyond that, discuss what to buy with your teammates.

Maybe an arty could pump economy as they farm so much (eg on hell's pass), but i would not gimp yourself by contributing more than your fair share.

depends on how you want to play really. the three techs I max out first are usually turret damage, turret range, and armor. next on the priority are firing rate and hull. nice to have but not important picks are hull crit, shields, scanner range.
but when going up against a team of guys that really like to go in close, range is obviously not that interesting.
and as for faction, whenever I'm done buying my top 3 important upgrades, we're still a ways away from the next tier, and I have CP to spare, then they go into faction, unless my income was low during the first two tiers, in which case I forego turret range in favor of advanced colonization

Yeah but the upgrades for 300 CP at the base won't do much there. However don't forget that your support ship offers some serious direct firepower, it has base stats on par with a cruiser and even more DPS from a full broadside at light turret range. The placeable turrets are an advantage but they are not your only weapon.

Yeah the support has a damn fine amount of firepower in its guns, so weapon upgrades are not a bad idea at all.

As far as fleet upgrades, you want to get by on the bare minimum you can.

You will want convoy and colinization asap (asuming you control at least one wormhole and one planet respectively) because they increase your cp income, and thus eventualy pay for themselves.
experimental armagedon missile is a good one of course, but thats becasue if the game is going late enouph to get it then it can make the difference between a win and a lose. Prototype is also nice as its an extra skill point, always a good thing.

but for the others, well, they are nice, but often not as nice as keeping your cannons and armor up to snuff, which results in you killing them more often (or getting away and denying them the CP).

And titans are worth a lot of cp, so they tend to help the enemy more than you, they ARE powerfull though, so if you are activly pushing on their base through the lines, then go ahead, but if its even a little bit back and forth, or middle fighty, then avoid that upgrade like the plauge.

Gun:
I go with Plasma for better farm and for finishing of guys hiding behind stuff. But its up to you really.

Crew:
Basically get all faction, hull, weapon discounts as soon as possible.

Ingame upgrades:

Weapon:Fire rate (max this)Weapon damage (max this)Shield critical (If you come up against lots of corvettes/frigattes this is a good upgrade to get late game)Weapon range (go this if you do not come up against a lot of corvettes/frigattes)

Hull:structure (max this)armor (if its a long game get this)

note:why structure instead of armor???
Structure gives you more effective health and negates the x3 dmg torpedoes do to armor.

If you find you are having problems with mines you can switch it out.
I go with Beacon on larger maps nowadays.

beacon at rank 3 gives a 30% range increase to all units inside its radius.
also turrets have a 50 unit fire range but only a 30 unit radar range.
so turrets combined with beacon is quite dangerous.
However, beacons have incredibly low health so you need to be smart on where to put em since they tend to get obliterated in the crossfire.

Also. an arillery unit in your team will make sweet love to you if you place a rank 3 beacon on him.

This makes for a nice wall of death to hide behind if you need to get away and repair.

Mines:

-Once you play a few games you will know where the enemy will try to sneak in to cap your lab.
3x mines will completely obliterate any tresspassers.

-also placing mines in nebulas where its likely the enemy will try to retreat to if their health gets low is also very effective.
Ive gotten TONNES of corvette/frigatte kills with this.
also hillarity ensues in chat afterwards.

-Farming, 1 mine will kill a full wave of creeps. effective if you find it problematic the use the mines in any other way.
Also, mines will ignore the force field.

Misc:

How do I counter artillery that keeps wreaking havoc on my turrets?

Well... I dont really have a good answer to this.
I find that artillery is my nemesis and the rock to my scissors.
If anyone has a good idea on this please let me know.

One thing you can think about is that some maps have artillery spots. where the bastard will ALWAYS be. place mines there
usually behind asteroid walls etc.

How do I counter Corvettes/frigattes?

These little gimps are incredibly deadly to you late game.
if you have gotten your farm on you should be able to invest in shield critical. This will make them sad pandas.

they will tend to Zip around you in a circle and just gnaw you down and then finsihing you of with a torpedo barrage.

If you dont have shield crit. place a mine just behind them and when they come around for the second pass BOOM. with their shields down they will either run away or die really fast.

-sorry for a messy post. I might clean it up a bit later -

EDIT:

Oh and regarding Fleet upgrades.
BIG NO NO!!!!!

Making your creeps stronger will make it harder for your team to farm cash since the battle line will move up towards the enemy base.

BUT!
There are 2 upgrades that you need to get as soon as humanly possible!
Convoy (if you own the portal)
Advanced colonisation (Muchos importantes)

Rockets and shields greatly increase the effect of creeps on PVP combat, they're worthwhile purchases if the PVP fights tend to stay close to the creep lanes so you can hide behind their shields and their rockets can dish out some serious pain to anyone trying to give chase. Torps only fire at turrets so only buy those once your creep waves are reaching the enemy turret lines.

The effect of mines and other cloaked things depends on the enemy scouting capabilities, a frig that loves its scanner (and it should since the scanner gives a damage bonus) will decloak any mines or beacons that are near combat zones.

Generally your turrets won't hold up to artillery fire, that's just how the game is. I found that the stealth generator at level 3 is decent at keeping arty from shelling your turrets since the arty can't just waltz in with a scanner and a level 3 stealth covers enough area that it can be placed far enough behind your line of defense that enemy scouts won't see it often. Of course the usual caveat about frigs with scanners applies here too.